On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

Abstract

In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

Numerical examination of concentration-dependent wastewater sludge ejected into a drinking water source

“…( 6), ( 7) will be presented based on bivariate spectral local linearisation approximation theory. Following Bellman and Kalaba 17 , including [18][19][20] we set Such that the quasi-linearized version of (6) becomes where the coefficients used in (9) are defined as It is well known that set of polynomials is dense in the set of continuous functions, therefore, to obtain the solution of the (6) subject to (7), we seek a series solution that is based on Lagrange cardinal polynomial, L p y L q (τ ) , approximation of the form:…”

“…( 6 ), ( 7 ) will be presented based on bivariate spectral local linearisation approximation theory. Following Bellman and Kalaba 17 , including 18 – 20 we set …”

Robustness of convergence demonstrated byparametric-guiding andcomplex-root-tunneling algorithms for Bratu’s problem

“…Following the principle of the spectral method (Muzara et al, 2018;Singh et al, 2020), we obtain four decoupled linear equations:…”

“…In other intervals, we can proceed with exactly the same argument. Case (b): We choose sin πx , sin 2 πx , sin3 πx , sin4 πx (orthogonal to one another) as basis functions such that an arbitrary curve e x − 1.7183 x − 1 within the interval [0, 1]. We acquire the residuals R ( x ) = e x − 1.7183 x − 1 − c 1 sin πx − c 2 sin 2 πx − c 3 sin3 πx − c 4 sin4 πx , where c 1 , c 2 , c 3 and c 4 denote coefficients that need to be found. Following the principle of the spectral method (Muzara et al , 2018; Singh et al , 2020), we obtain four decoupled linear equations: After computations, we obtain c 1 = − 0.2178 , c 2 = 0.0135, c 3 = −0.0088 and c 4 = 0.0017. Computed and exact curve are observed to be nearly identical. If more basis functions are chosen, the accuracy of the approximation can only become higher expectedly.…”

“…Following the principle of the spectral method (Muzara et al , 2018; Singh et al , 2020), we obtain four decoupled linear equations: After computations, we obtain c 1 = − 0.2178 , c 2 = 0.0135, c 3 = −0.0088 and c 4 = 0.0017.…”